Many-Particle Li Ion Dynamics in LiMPO4 Olivine Phosphates (M = Mn, Fe)

LiMPO4 (M = Mn, Fe) olivine phosphates are important materials for battery applications due to their stability, safety, and reliable recharge cycle. Despite continuous experimental and computational investigations, several aspects of these materials remain challenging, including conductivity dimensionality and how it maps onto Li pathways. In this work, we use a refined version of our finite temperature molecular dynamics “shooting” approach, originally designed to enhance Li hopping probability. We perform a comparative analysis of ion mobility in both materials, focused on many-particle effects. Therein, we identify main [010] diffusion channels, as well as means of interchannel couplings, in the form of Li lateral [001] hopping, which markedly impact the overall mobility efficiency as measured by self-diffusion coefficients. This clearly supports the need of many-particle approaches for reliable mechanistic investigations and for battery materials benchmarking due to the complex nature of the diffusion and transport mechanisms.


S1.1 Kinetic Energy Redistribution and System Response
The shooter move selectively warms up the Li + ions by transferring a variable amount of kinetic energy from the host framework (MPO4, M = Fe/Mn) to the mobile ions (Li + ). This way a clear separation of velocities, and hence kinetic energy, can be achieved. This separation is explicitly calculated in Figs. Figure S1: The effect of one 'Shooting' event on the total kinetic energies of each constituent atom type in LiFePO4 (top) and the effect of multiple 'Shooting' events (bottom).
The application of the 'Shooter' algorithm generates a separation in kinetic energy between Li + and FePO4 (Fig. S7, upper part). This separation is attained via the generation of a velocity S4 distribution not typical of the ensemble. This separation gradually decays and after approximately 0.5 ps the equilibrium distribution is recovered, up to some echoes between 1.0 and 2.0 ps from the pulse. Multiple applications of the 'Shooter' algorithm, every 0.5 ps as shown in Fig. S7 (lower part), establishes and partially maintains a separation, keeping therefore some amount of bias on the distribution. The same effect is achieved in applying the 'Shooter' algorithm to LiMnPO4, as shown in Fig. S8. Figure S2: The effect of one 'Shooting' event on the total kinetic energies and their distribution on each constituent atom type in LiMnPO4 (top) and the effect of multiple 'Shooting' events (bottom). Atoms are color-coded.

S1.2 Velocity Autocorrelation Function of Li + in LiFePO4 and LiMnPO4
Figure S3: Normalized velocity autocorrelation function of Li + ions within LiFePO4, (top) and LiMnPO4 (bottom) for both, a MD simulation of an equilibrated structure at 700 K (left, red) and a resulting velocity distribution after a 'shooting' event (right, blue). The total simulation time was 10 ps, 10 VACF functions of 1 ps each were calculated and averaged.
From the velocity autocorrelation function of Li + ions in Figure S3, a time interval of 0.5 picoseconds between two successive shooting moves is enough to allow system equilibration and decorrelation of Li + dynamics. To understand how the shooter approach acts on the system equilibrium distributions, the VACF of equilibrium velocity distributions are shown in Fig. S3 and compared to post-'shooting' event velocity distributions for LiMPO4. In both cases an initial rapid decay is observed, followed by damped oscillations around zero. After about 0.5 ps, the velocities are largely decorrelated, in both straightforward and biased simulation regimes. The VACF autocorrelation functions indicate that the shooter-perturbed system returns to equilibrium in a way that is indistinguishable from its response to spontaneous fluctuations, in line with Onsager's regression hypothesis. Our approach therefore excites specific fluctuations, while (in principle) preserving the "measurability" of a system close enough to equilibrium.

S1.3 Methodology
As its name suggests, this approach is a modification of the shooter algorithm used in transition path sampling, which is employed here to systematically perturbate points in phase space 33 . The following general steps describe a typical 'Shooter' move simulation used throughout this body of work: i. A thermally equilibrated configuration was used to initialize the simulation at 700 K. ii.
A small perturbation was chosen to Li ions only, by setting a Gaussian half-width, centered on the velocity of each Li ions. An initial half-width was chosen in two different simulations regimes, as described below.
iii. The instantaneous frame temperature was calculated from the total kinetic energy, . Here, the sum is calculated over all atoms of mass and velocity . is the Boltzmann constant and is the number of degrees of freedom. iv.
Velocities of Li ions are perturbed by choosing a random value from the Gaussian distribution centered around its current velocity. The new velocity distribution is A post-perturbation temperature was calculated from the rescale kinetic energy. vi.
The initial temperature was restored through rescaling the velocities of all other particles by = √ .
vii. The frame is propagated for a set amount of time (at least 0.5 ps as indicated by the analysis of the vac(t), see main text for details).
viii. The Gaussian half-width is increased by a smearing factor, typically 1.0001.
ix. This process is repeated until the total simulation time reaches a target value of choice.
Typical simulation times were in the order of 300-500 ps for mechanistic assessments and 3-5 ns for MSD/Diffusion Constants evaluations, as described below.

S7
The implementation of the shooter-enhanced MD simulation was achieved as chain of shooting steps, regularly spaced in time. The VACF relaxation time served as a lower relaxation limit for the time delay between shooting events, at least 0.5 ps. The Gaussian half-width, which controls the extent of kinetic energy transfer between host framework and Li ions, was chosen within the interval [5.10 -5 -10 -2 Å/fs], the limits corresponding to a low and high shooting regime, respectively. Most simulations (unless otherwise indicated) were performed using a longer delay of 2 ps between shooting events. This choice was motivated by kinetic energy distribution analysis after the application of both single and sequential shooting pulses. A single pulse (Figs. S1 and S2) causes echoes in the kinetic energy fluctuations of heavier particle after 0.5 ps from shooter onset. If too closely sequenced (0.5 ps), shooting pulses would enforce specific kinetic energy separations, by partially preventing system relaxation. The choice of 2 ps represents a good compromise in keeping as close as possible to equilibrium distributions, but not too close to quench diffusive behavior. Clearly, any attempt to "optimize" shooting moves must be understood as an effort to keep the shooting perturbations as small as possible (and therefore as infrequent as possible), while maintaining steady particle motion in a linear regime, where Kubo-Green measurements can be performed. Accordingly, the aim of the shooting approach is to amplify particle translocation probability and many-particle propagation within materials as a basis for a detailed mechanistic analysis. Achieving a constant rate of hopping is instrumental to calculating diffusion coefficients based on the Einstein relation, which relates the diffusion constant to the slope of mean square displacement (MSD) as a function of time, in the long-time limit. Calculations of diffusion constants for different choices of the shooting parameters allowed in turn to assess numerical stability of the overall shooter MD approach to Li + ion dynamics.

S3.1 LiFePO4 -Low Shooting Regime
Figure S10: Snapshots of a representative sequence mechanism along [010] in LiFePO4. All Li + are individually colored (periodic images are same-colored). The channel is first activated by the formation of a Frenkel defect (a), which occurs via the combination of multiple Li + ion jumps. Vacancy (square) and double occupancy begin to migrate down the channel via single Li + ion jumps (b, c) until recombination (e). As a result, the column of Li + ions have moved down one crystallographic site (e). Figure S14: LiFePO4 (low). MSD vs. time for all Li + ions before the cross-jump event (black) and after (grey). Ds = 1.03 ± 0.08 × 10 −8 2 −1 (before) and Ds = 6.45 ± 0.20 × 10 −8 2 −1 (after). S15

S5 Tight-Binding Molecular Dynamics
Constrained NVT MD simulations were run based on a collective variable (CV), represented by the distance of a Li + ion from a "plane" cutting through [010] channels, defined from 3 M atoms (either Fe or Mn) surrounding the channel. A single Li + ion was restrained by a harmonic potential between two interstitial sites, "above" and "below" the plane. The resulting sequence of ion displacements is summarized below for LiFePO4 and LiMnPO4. Force calculations were based on a transferable, semiempirical third-order tight-bind potential, GFN-xTB [S1] as implemented in CP2K. As a variant development of DFTB3 [S2], xTB allows for the description of complex systems containing several interaction types, including covalent, ionic, and dispersive interactions. As a reactive potential, bond-breaking and complicated atomic rearrangements can be reliably and efficiently accounted for. The integration step was 0.2 fs, the temperature of the Nose thermostat was chosen identical to the shooter simulations, 700 K. Electrostatic interactions were accounted for with an Ewald summation, while D3 dispersion correction was used. Slater orbitals were expanded in GTO (Gaussian Type Orbitals) up to sixth order. All other parameters were left at their default values. The simulation box contained 168 atoms (24 Li ions).

S5.1 LiFePO4
Figure S20: Sequence of [010] in-channel displacements in LiFePO4 in response to the shift of a single Li + ion, grey atom in a). The formation of Frenkel defects followed by Li + ion S19 rearrangements (b-e) leads to an overall displacement of Li + ions by one site, "upwards". Li ions are individually colored (same color for periodic images).

S5.2 LiMnPO4
Figure S22: Sequence of [010] in-channel displacements in LiMnPO4, in response to the bias of a single Li + ion, lowest grey atom in a). The formation of Frenkel defects (b) followed by Li + ion displacement sequences (b-e) leads to an overall displacement of the Li + ion by one site, "upwards", f). Li ions are individually colored (same color for periodic images).